I will describe a recently-developed approach to studying experimental data obtained from a periodic or quasiperiodic dynamical system. The idea is to use persistent cohomology to construct robust new coordinate functions on the data. These coordinates take values in the circle, so they may correspond to phase variables or angular variables implicit in the system being studied. This is joint work with Primoz Skraba and Mikael Vejdemo-Johansson, with a key technical contribution by Dmitriy Morozov.